To obtain curve crossing and curve veering k23 e k31 have been used as parameters. This allows the system to lose symmetry property, and desired phenomena to set up. The different behaviour is described through the following two cases: k23 = 1 N/m curve crossing (coincident eigenvalues), k23 = 1.02 N/m curve veering (close eigenvalues). It is straightforward that crossing and veering phenomena are tightly linked with certain physical parameters of the system. This last concept is used to design the test rig. Solving the eigenvalue problem for each value of k31 and plotting eigenvalues curves of mode 2 and mode 3 versus this parameter, leads to the results displayed in Figure 2. In the first case of perfect cyclic and symmetric system (k23 = 1 N/m), applying MAC between eigenvectors close to the crossing point (k31 = 0.99 N/m and k31 = 1.01 N/m) it possible to easily depict the mode reversal (Figure 3 on the left). During curve crossing phenomena, involved modes maintain their peculiarities, but they do swap: mode 2 become mode 3 and viceversa. Orthogonality of eigenvectors is guaranteed, due to mass matrix proportional to identity. This change is only a marginal effect because order is only linked with frequency value and not with substantial mode characteristics. This is confirmed by observing eigenvectors before and after occurrence of crossing phenomena in Figure 4 on the left: every mode maintains its property and changes only the identification number. In the exact configuration of crossing, system shows two coincident eigenvalues and the associated eigenspace has size two, while in all other cases, where eigenvalues are very close but not coincident, each eigenspace has size one. In the second case of non perfect symmetric system (k23 = 1.02 N/m), solving the eigenvalue problem for each value of k31 and plotting eigenvalues curves of mode 2 and mode 3 versus this parameter, leads to the results displayed in Figure 2 on the right. Starting from k31 = 0.9 N/m and by increasing this stiffness value, eigenvalues curves get closer but they abruptly veer away. In this case there are not coincident eigenvalues, not even for k31 = 1. Applying MAC for two different values of k31 (k31 = 0.99 N/m) and after (k31 = 1.01 N/m), where the distance between the two eigenvalues is minimum, it is possible to obtain the result reported on the right of Figure 3. It is straightforward to understand that the modes undergo a modal properties change, maintaining the initial rank. The modal assurance criterion confirms the alteration of mode shape by varying one of the parameter of the system. This result agrees to the Leissa’s sentence [1], in fact the modes shapes across the curve veering mix their dynamic properties in a continuous manner. The strange and sudden variation of the 2nd and 3rd mode shapes can be seen in Figure 4 on the right. Presence of symmetry could be justify because results come from physical system with symmetry and cyclic properties, despite of the abrupt change is very clear. Leissa said that was very difficult to identify a continuity between modes shapes [1], as confirmed in Figure 4. In fact, despite the step of k31 has been refined, modes shapes suddenly change. 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 Curve Crossing: k 12 = 1 N/m; k 23 = 1 N/m k 31 [N/m] Eigenvalues [rad2/s2] 2 3 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 Curve Veering: k 12 = 1 N/m; k 23 = 1.02 N/m k 31 [N/m] Eigenvalues [rad2/s2] 2 3 Figure 2 – Natural frequencies of the lumped system versus k31 parameter: curve crossing with k23 = 1 N/m (left), curve veering with k23 = 1.02 N/m (right). 328
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